We first notice that since \({w}\)  are non-increasing, the rearrangement inequalities [1]} gives:
\(h({z}) = -\min \limits _{\sigma \in \mathfrak {S}_n} {w_\sigma ^\intercal }{z}= \max \limits _{\sigma \in \mathfrak {S}_n} -{w_\sigma ^\intercal }{z}.\) 
Thus, \(h\)  is the support function of the convex set \(\mathcal {C}({\tilde{w}})\) , since:
\(h({z}) = \max _{\sigma \in \mathfrak {S}_n} -{w_\sigma ^\intercal }{z}= \sup _{{y}\in \mathcal {C}({\tilde{w}})} {y}^\intercal {z}.\) 
